2 Answers
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There are sources beginning in the 1200s (the Mordechai and others) that quote the practice to avoid various legumes and semigrains; either because of concern that they make contain some wheat (or other chametz-causing grain) mixed in; or because if you grind them into flour, people may think you're using wheat flour (or barley spelt etc.) and making chametz. See this question for more details. Rabbi Joseph Soloveichik famously observed that a Rabbi Yohanan ben Nuri had prohibited rice in the Talmud; thus the custom has at least some minority Talmudic opinion on which to hang its hook (even though for entirely different reasons).

The custom against kitniyot is codified as Ashkenazic practice by Rabbi Moshe Issreles in his comments on the Shulchan Aruch (c. 1550).

As it has become established custom, it is now binding on Ashkenazic Jews, barring rare circumstances (someone could ask their rabbi for a waiver if they have severe dietary restrictions; in times of severe scarceness, some communities' rabbinic leadership would allow some forms of kitniyot that year). Rabbi Danczig, writing in the Chayei Adam right about 1800, explains that it was not instituted top-down by rabbinic leadership (in which case it would warrant the Mosaic prohibition "do not deviate from what your judges state, left or right"), but something that happened bottom-up, in a softer, more organic fashion, and has now been fixed. He applies the opening verse of Proverbs, "don't reject the teachings of your mother." Around the year 1900, Rabbi Yechiel Michel Epstein of Novardok (in Aruch HaShulchan) wrote that "those Ashkenazic Jews who fail to keep kitniyot today, it is clear that they lack all fear of Heaven."

We view standardized custom as something very strong and not something that should be tinkered with. We don't think we're so smart that 800 years of Jewish history can just be thrown out the door; some caution and humility are in order.

There are, however, a few loopholes:

The custom clearly never applied to Sephardic Jews. There are different opinions regarding whether you can observe the Sephardic custom if you move to a Sephardic place (with Rabbi Ovadiah Yosef shlit'a of the opinion that the entire Israel is "a Sephardic place" and thus the lenient custom prevails there; this argument would effectively do away with Ashkenazic custom wholesale), or according to some, even if you're visiting a Sephardic home. There are also Jews who have changed over their entire set of customs consistently to a different community's; please, consult your rabbi first.

Rabbi Moshe Feinstein's opinion was that kitniyot only covers those legumes/semigrains explicitly banned at some point in time (some back in the 1200s, others later, as evidenced by New-World crops such as corn being on the list). If it hasn't been banned yet, it's not prohibited (unless you want to go make a ban on it now, and he suspects that rabbis stopped adding to the prohibited list at some point to make sure there were enough edible substances remaining! This is Rabbi Feinstein projecting his considerate, nuanced method of psak, by the way.) This approach has the added convenience of sparing you the mental gymnastics required to explain why various products did or didn't make the list -- the list happened as it happened over time.

A body with the force of a Sanhedrin would certainly have the power to repeal this custom, and I have heard Ashkenazim grumble and hope that happens soon. The Sanhedrin recently established in Israel (using Rabbi Halbershtam as its nexus, as was done with Rabbi Beirav 500 years ago) is cautious not to overstep its boundaries so as not to be ignored entirely (many don't accept it as is), so I don't see them as doing this anytime soon.

As per Rav Moshe Feinstein's approach, that the prohibition to eat kitniyos is a hanhagah shehinhugu chachamim, a custom that the Rabbis guided their flocks to keep. It does not have the status of a takannah, a law instituted by court. All the more so is this true according to the Chayei Adam's approach mentioned by Shalom.

Those Ashkenazi gedolim who did not like the custom lived in times when it was either not unanimously accepted yet in the Ashkenazic world and they wished to keep it that way (some Rishonim), or it was recently accepted by a previous generation and the gadol was expressing frustration that he could not do away with it (Rav Yakov Emden).

in R' Moshe's responsum on peanuts he suggests only as a possibility that it was a recommendation and not an enactment. (Is there a different responsum where he says that definitively?) If we assume it's a neder, how can someone sick get a dispensation? And why is a minor quantity of kitniyot batel; if I made a neder not to eat corn, it would require 60x more or less.
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ShalomJul 29 '11 at 0:54

R' Moshe actually is matir peanuts on Pesach (except in communities that already refrain from eating them) based on what you refer to as a "suggestion of a possibility". Where in his teshuvah do you see him only suggesting it as a possibility? The mistake most people make is from the footnotes to the Semak (Amudei Golah) in which it is written that that kitniyos is "Mita'am gezeirah". Many read that as if kitnoyos is a gezeirah.
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YahuJul 29 '11 at 14:50

The point is that it is not a gezeirah; rather it is a minhag that was accepted in order to be a practical fence (which is the idea behind gezeiros) to prevent us from mistakes regarding the use of chametz on Pesach. The taste of kitniyos was not what they accepted as a neder. They accepted to refrain from eating kitnoyos on Pesach. According to your approach to nedarim, how would you explain the fact that hanodeir min hahalav mutar beta'aroves? As long as the milk is not apparent it is permissible even with less than 60x more!
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YahuJul 29 '11 at 14:55

As far as someone who is sick goes, the custom was never accepted that someone who is sick should refrain from eating kitnoyos on Pesach in the same way that it was never accepted as a custom for times of famine or regarding any other benefit other than eating. The idea that the custom was accepted with limitations to how stringent we go with it is not a contradiction to the fact that breaking with the custom has the severity of breaking a neder.
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YahuJul 29 '11 at 14:59